A GENERAL FORM OF JORDAN'S INEQUALITIES AND ITS APPLICATIONS

作者：Zhu Ling^*

来源：Mathematical Inequalities & Applications, 2008, 11(4): 655-665.

DOI：10.7153/mia-11-55

• Lower and upper bounds
• Jordan's inequalities
• spherical Bessel functions
• infinite series
• Kober's double inequality

In this work. the general form of Jordan's double inequalities:
P-2N(x) + alpha(r(2) - x(2))(N+1) <= sin x/x <= P-2N(x) + beta(r(2) - r(2))(N+1)
are established, where x is an element of (0,r],r <= pi/2, P-2N(x) = Sigma(N)(n=0) a(n)(r(2) - x(2))(n), a(0) = sin r/r, a(1) = sin r-r cos r/2r(3), a(n+1) = 2n+1/2(n+1)r(2)a(n) - 1/4n(n+1)r(2)a(n-1) , N >= 0 is a natural number, alpha = a(N+1) and beta = 1 - Sigma(N)(n=0) a(n)r(2n)/r(2(N+1)) are the best constants in inequalities above. The application of the results above give a new infinite series (sin x)/x Sigma(infinity)(n=0)a(n)(r(2)-x(2))(n) for 0 < vertical bar x vertical bar <= r <= pi/2, the general improvement of Yang Le inequality, and a ageneral form a kober's double inequality.

• 出版日期2008-10
• 单位浙江工商大学